On the mean Euler characteristic of contact manifolds (Q2874722)

The mean Euler characteristic of a contact manifold is defined as an average of the alternating sums of the dimensions of the contact homology groups. It is assumed that these dimensions are only infinite in a finite range of degrees, which is excluded from the sum. NEWLINENEWLINENEWLINEThe first result in this paper is a generalisation of a result in [\textit{V. L.\ Ginzburg} and \textit{E.\ Kerman}, Int. Math. Res. Not. 2010, No. 1, 53--68 (2010; Zbl 1254.53113)]. In that paper, a formula for the mean Euler characteristic is given in cases where the flow of the Reeb vector field has finitely many closed simple orbits. In the paper under review, this condition is relaxed to an \textit{asymptotic finiteness} condition. This notion involves a sequence of contact forms \((\alpha_r)_r\) for the given contact structure, such that all these contact forms admit a set of simple Reeb orbits, whose mean indices converge as \(r \to \infty\). Under this condition, the author proves an expression for the mean Euler characteristic in terms of data on ``good'' principal orbits. Under additional assumptions, there is an Morse-Bott type expression for the mean Euler characteristic in terms of maximal orbifolds. NEWLINENEWLINENEWLINEThe author also investigates the mean Euler characteristic of a contact manifold obtained from an asymptotically finite contact manifold by subcritical contact surgery. In dimension at least 3, and if there are no Reeb orbits of degree \(-1, 0\) or 1, the mean Euler characteristics of the two contact manifolds differ by \(\pm 1/2\).